Contribution of the spin-zeeman term to the binding energy for hydrogen in non-relativistic QED.

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1 Annals of the University of Bucharest (mathematical series) 4 (LXII) (2013), Contribution of the spin-zeeman term to the binding energy for hydrogen in non-relativistic QED. Jean-Marie Barbaroux and Semjon Vugalter Abstract - We show that the spin-zeeman term contributes at least to the same order as the first radiative correction of the binding energy for hydrogen atom in non relativistic quantum electrodynamics obtained in the spinless case [8]. Key words and phrases : ground state energy. Pauli-Fierz Hamiltonian, binding energy, Primary 81Q10; Sec- Mathematics Subject Classification (2010) : ondary 35P15, 46N50, 81T Introduction For a hydrogen-like atom consisting of an electron interacting with a static nucleus of charge ez described by the Schrödinger-Coulomb Hamiltonian αz/ x, the quantity inf spec( ) inf spec( αz x ) = (Zα)2 4 corresponds to the binding energy necessary to remove the electron to spatial infinity. The interaction of the electron with the quantized electromagnetic field is accounted for by adding to αz/ x the photon field energy operator H f, and an operator I(α) which describes the coupling of the electron to the quantized electromagnetic field, yielding the so-called Pauli-Fierz operator (see details in Section 2). In this case, the binding energy is given by Σ 0 Σ := inf spec ( + H f + I(α) ) inf spec ( αz x + H f + I(α) ) (1.1) The free infraparticle binds a larger quantity of low-energetic photons than the confined particle and thus possesses a larger effective mass. In order for the particle to leave the potential well, an additional energetic effort is 307,

2 308 Jean-Marie Barbaroux and Semjon Vugalter therefore necessitated compared to the situation without coupling to the quantized electromagnetic field. It remains a difficult task, however, to determine the binding energy. There are mainly two difficulties. The first is that the ground state energy is not an isolated eigenvalue of the Hamiltonian, and can not be determined with ordinary perturbation theory. The second is due to the infrared problem in quantum electrodynamics whose origin is in the photon form factor in the quantized electromagnetic vector potential occurring in the interaction term I(α), that contains a critical frequency space singularity. The systematic study of the Pauli-Fierz operator, in a more general case involving more than one electron, was initiated by Bach, Fröhlich and Sigal [3, 4, 5]. Later on, several rigorous results [15, 18, 16, 12, 14, 6, 17, 2, 7, 8, 10, 19, 11] have been obtained addressing both qualitative and quantitative estimates on the binding energy and the ground state energies Σ 0 and Σ occuring in (1.1). The case of spinless particle attracted most of the attention since the additional spin-zeeman term ασ B(x) in the case of a spin 1/2 particle induces substantial technical difficulties for quantitative estimates. From a rigorous point of view, if one takes into account the spin of the particle, it is not clear of what order the first correction in powers of the fine structure constant α is. This question is sensible since both the self-energy Σ 0 and the ground state energy Σ for Hydrogen atom, up to a normal ordering constant, are of the order α 2 in the case of a spinless particle (see [7, 8, 10, 17] and references therein), whereas in the case of an electron, i.e. a spin 1/2 particle, they are proportional to α (see e.g. [16, 12, 11] and references therein). In the latter case, though, the binding energy is still expected to be proportional to α 2 in the leading order. This fact together with the upper bound on the contribution of the spin-zeeman term to the binding energy will be proved by the authors in a subsequent paper; see also Remark 2.2. In the present paper, we prove Theorem 2.1 which gives a lower bound on this contribution. 2. Model and main result We study an electron, i.e., a spin 1/2 particle, interacting with the quantized electromagnetic field in the Coulomb gauge, and with the electrostatic potential generated by a nucleus. The Hilbert space accounting for the Schrödinger electron is given by H el = L 2 (R 3 ) C 2. Here R 3 is the configuration space of the particle, while C 2 accommodates its spin.

3 Contribution of the spin-zeeman term The Fock space of photon states is given by F = n N F n, where the 0-photon space is F 0 = C, and for n 1 the n-photon space F n = n ( s L 2 (R 3 ) C 2) is the symmetric tensor product of n copies of onephoton Hilbert spaces L 2 (R 3 ) C 2. The factor C 2 accounts for the two independent transversal polarizations of the photon. On F, we introduce creation and annihilation operators a λ (k), a λ(k) satisfying the distributional commutation relations [ a λ (k), a λ (k ) ] = δ λ,λ δ(k k ), [ a λ (k), a λ (k ) ] = [ a λ (k), a λ (k ) ] = 0. There exists a unique unit ray Ω f F, the Fock vacuum, which satisfies a λ (k) Ω f = 0 for all k R 3 and λ {1, 2}. The Hilbert space of states of the system consisting of both the electron and the radiation field is given by H = H el F. We use atomic units such that = c = 1, and where the mass of the electron equals m = 1/2. The electron charge is then given by e = α, where the fine structure constant α has physical value about 1/137 and will here be considered as a small parameter. Similarly to the Pauli operator which acts on Hilbert space L 2 (R 3, C 2 ) and describes the energy of a spin 1/2 particle interacting with classical external magnetic field, the Pauli-Fierz operator we consider in this paper is the Hamiltonian for a particle interacting with the quantized radiation field (see [3, 4, 5] and references therein). For an atom with nuclear charge Z = 1, this operator is defined by : ( i x I f + αa(x) ) 2 : + α σ B(x) + V (x) If + I el H f, (2.1) where V is the electrostatic potential. The operator that couples a particle to the quantized vector potential is A(x) = A (x) + A + (x), where A (x) = λ=1,2 A + (x) = λ=1,2 R 3 R 3 ζ( k ) 2π k 1/2 ε λ(k)e ikx a λ (k)dk, ζ( k ) 2π k 1/2 ε λ(k)e ikx a λ (k)dk,

4 310 Jean-Marie Barbaroux and Semjon Vugalter and where diva = 0 by the Coulomb gauge condition. The vectors ε λ (k) R 3 (λ = 1, 2), are the two orthonormal polarization vectors perpendicular to k, ε 1 (k) = (k 2, k 1, 0) k k 2 2 and ε 2 (k) = k k ε 1(k). The function ζ( k ) implements an ultraviolet cutoff, independent of α, on the photon momentum k. We assume ζ to be of class C 1 and to have a compact support. The symbol :... : denotes normal ordering and is applied to the operator A(x) 2. It corresponds here to the subtraction of a constant operator c n.o. α, with c n.o. = [A (x), A + (x)] = (2/π) 0 r ζ(r) 2 dr. The operator that couples a particle to the magnetic field B = curla is given by B(x) = B (x) + B + (x), where B (x) = ζ( k ) λ=1,2 R 3 2π k 1/2 k iε λ(k)e ikx a λ (k)dk, B + (x) = ζ( k ) 2π k 1/2 k iε λ(k)e ikx a λ (k)dk. λ=1,2 R 3 In Equation (2.1), σ = (σ 1, σ 2, σ 3 ) is the 3-component vector of Pauli matrices ( ) ( ) ( ) i 1 0 σ 1 =, σ =, σ i 0 3 =. 0 1 The Coulomb potential is the operator of multiplication by V (x) = α x. The photon field energy operator H f is given by H f = λ=1,2 R 3 k a λ (k)a λ(k)dk. In the sequel, instead of the operator (2.1), we shall proceed to a change of variables, and study the unitarily equivalent Hamiltonian ( H = U : ( i x I f αa(x) ) 2 : + ασ B(x) + V (x) If + I el H f )U, (2.2)

5 Contribution of the spin-zeeman term where the unitary transform U is defined by and P f = λ=1,2 U = e ip f.x, k a λ (k)a λ(k)dk is the photon momentum operator. We have Ui x U = i x + P f, UA(x)U = A(0), and UB(x)U = B(0). In addition, the Coulomb operator V, the photon field energy H f, and the photon momentum P f remain unchanged under the action of U. Therefore, in this new system of variables, and omitting by abuse of notations the operators I el and I f, the Hamiltonian (2.2) reads H = : ( (i x P f ) αa(0) ) 2 : + ασ B(0) α x + H f, (2.3) where :... : denotes again the normal ordering. The Hamiltonian for a free electron, i.e., a free spin 1/2 particle, coupled to the quantized radiation field is given by the self-energy operator T, T = H α x = : ( (i x P f ) αa(0) ) 2 : + ασ B(0) + Hf, (2.4) where we omit again the operators I el and I f. This system is translationally invariant, that is, T commutes with the operator of total momentum P tot = p el + P f, where p el and P f denote respectively the electron and the photon momentum operators. Therefore, for fixed value p R 3 of the total momentum, the restriction of T to the fibre space C 2 F is given by (see e.g. [13, 11]) T (p) = : (p P f αa(0)) 2 : + ασ B(0) + H f. (2.5) Henceforth, we will write A ± = A ± (0) and B ± = B ± (0). The ground state energies of T and H are respectively denoted by Σ 0 = inf spec(t ) and Σ = inf spec(h),

6 312 Jean-Marie Barbaroux and Semjon Vugalter and the binding energy is defined by Σ 0 Σ. It is proven in [1, 13] that Σ 0 = inf spec(t (0)), and Σ 0 is an eigenvalue of the operator T (0). Our main result is the following, Theorem 2.1. The binding energy fulfills the following inequality where Σ 0 Σ 1 4 α2 + ( ) e (1) + e (1) Zeeman α 3 + O(α 4 log α ), (2.6) e (1) = 2 ζ 2 (t) dt and e(1) Zeeman 3π t = 2 t 2 ζ 2 (t) 3π 0 (1 + t) 3 dt. Remark 2.1. We recall that for the spinless Pauli-Fierz model it is known (see [8] and references therein) that the binding energy is Σ 0 Σ = 1 4 α2 + e (1) α 3 + O(α 4 ). The above Theorem 2.1 thus shows that the spin-zeeman term yields an additional contribution of order at least α 3. Remark 2.2. In a forthcoming paper we will show that this result is optimal, namely that the inequality (2.6) can be turned into an equality, by deriving a sharp upper bound for the binding energy up to the order α 3. This upper bound will coincide with the lower bound of the work at hand and gives the correct coefficient of the order α 3 in the expansion of the binding energy in powers of α. Such an estimate is much more involved than the proof of the lower bound which requires only a construction of a trial function. In addition to the problems encountered in the spinless case, there are several additional difficulties when taking into account the spin-zeeman term. The degeneracy of the ground state in the spin case (see [21, 20] and references therein) gives rise to technical difficulties. A more severe problem for the proof of the upper bound is that the ground state energy Σ 0 of the self-energy operator T given by (2.4) is of the order α and not of the order α 2 as in the spinless case ([11]). In addition, the photon number bound for a ground state of H, which is a crucial estimate for the proof of the upper bound, is only of the order α, instead of α 2 in the spinless case.

7 Contribution of the spin-zeeman term In the remainder, we will need the following notations. For n N, let Π n be the orthogonal projection onto the subspace H el F n of the ( space H el F, ) and Π n be the orthogonal projection onto the space H el k n F k. On H el F, we define the positive bilinear form v, w := v, (H f + P 2 f )w, and its associated semi-norm v = v, v. Proof. To prove the theorem we will construct a trial function Ψ trial such that holds Ψ trial, HΨ trial / Ψ trial 2 Σ 0 α 2 /4 (e (1) + e (1) Zeeman )α3 + O(α 4 log α ). Let P := i x. We denote by θ GS the ground ( state ) of T (0) with the normalization condition Π 0 θ GS = Ω f, where = is the normalized spin up component 1 0 (see(a.3)-(a.4) in Theorem A.1 for detailed definiton and properties of θ GS ), and let Θ := u α θ GS For Γ 1 defined as in (A.1) by Γ 1 := (H f + P 2 f ) 1 σ B + Ω f, and u α the normalized ground state of the Schrödinger operator α/ x, we set and u α = 1 8π α 3/2 e α x /2. (2.7) Φ α := 2 P P f (H f + P 2 f ) 1 u α Γ 1, (2.8) Υ α := 2 χ (α, ) (H f )(H f + P 2 f ) 1 P A + u α Ω f, (2.9) where χ (α, ) (H f ) is an infrared cutoff and χ (α, ) is the characteristic function of (α, ). Let us define the following trial function Ψ trial = Θ + α 1 2 Φα + α 1 2 Υα. The state Ψ trial has only non zero vacuum and one-photon component, i.e., Π 2 Ψ trial = 0. In comparison with the trial function used in the spinless case [8] to recover the estimate up to the order α 3, with error α 4, the function Ψ trial differs

8 314 Jean-Marie Barbaroux and Semjon Vugalter in two points. First we pick now the state θ GS as the ground state of the translation invariant operator T (0) with spin. Second, we have an additional vector Φ α at the origin of the e (1) Zeeman α3 term in (2.6). From the definition of H, expanding (2.3) and taking into account the normal ordering, we obtain H =( α x ) + (H f + Pf 2 ) 2 P P f 4α 1 2 Re P A (2.10) + 4α 1 2 Pf A + 2αA + A + 2α(Re A ) 2 + 2α 1 2 Re σ B We shall use this expression to estimate all terms occurring in Ψ trial, H Ψ trial ) = Θ + α 1 2 Φα + α 1 2 Υα, H (Θ + α 1 2 Φα + α 1 2 Υα. (2.11) Step 1. We first compute the direct terms Θ, HΘ, α 1 2 Φ α, Hα 1 2 Φ α and α 1 2 Υ α, Hα 1 2 Υ α. Since θ GS is a ground state vector of T (0), and using orthogonality between the components of P u α and u α, we have Θ, HΘ = u α θ GS, H u α θ GS = u α 2 θ GS, T (0) θ GS + θ GS 2 u α, ( α x ) u α (2.12) = (Σ 0 α2 4 ) Θ 2. Using that for i, j, k {1, 2, 3}, u α / x i and 2 u α /( x j x k ) are orthogonal, and the fact that non particle conserving operators have mean value zero in the state Φ α, yields α 1 2 Φα, Hα 1 2 Φα ( = α 1 2 Φα, α x = α Φ α 2 + O(α 4 ), + (H f + P 2 f ) + αa A + ) α 1 2 Φα (2.13) where the last inequality holds since P u α = O(α), ( α x )P u α = O(α 3 ). Using the same arguments as above, and the fact that (H f + P 2 f ) 1 χ (α, ) (H f )(A + ) j Ω f = O( log α 1 2 ), the last direct term can be estimated as α 1 2 Υα, Hα 1 2 Υα ( = α 1 2 Υα, α ) x + (H f + Pf 2 ) + αa A + α 12 Υα = O(α 5 log α) + α Υ α, (H f + P 2 f )Υ α + O(α 4 ) = α Υ α 2 + O(α 4 log α ). (2.14)

9 Contribution of the spin-zeeman term Step 2. We compute in (2.11) the cross terms with Φ α and Υ α. Using as above the estimates (H f + P 2 f ) 1 χ (α, ) (H f )(A + ) j Ω f = O( log α 1 2 ), P u α = O(α), and ( α x )P u α = O(α 3 ) yields α 1 2 Φα, ( α x )α 1 2 Υα + α 1 2 Υα, ( α x )α 1 2 Φα = O(α 5 log α 1 2 ). Due to Lemma B.1 (see Appendix B) holds (2.15) α 1 2 Φα, (H f + P 2 f )α 1 2 Υα + α 1 2 Υα, (H f + P 2 f )α 1 2 Φα = 0. (2.16) Furthermore, A Φ α c Φ α = O(α) and A Υ α c H 1 2 f Υ α = O(α) implies α 1 2 Φα, 2αA + A α 1 2 Υα + α 1 2 Υα, 2αA + A α 1 2 Φα 4α 2 A Φ α A Υ α = O(α 4 ). (2.17) In addition, due either to the symmetry of u α or the occurrence of nonparticle conserving terms, all other cross terms with Φ α and Υ α in (2.11) are equal to zero. Therefore, collecting (2.15)-(2.17) we get α 1 2 Φα, H α 1 2 Υα + α 1 2 Υα, H α 1 2 Φα = O(α 4 ). (2.18) Step 3. We estimate in (2.11) the cross terms involving Φ α and Θ = θ GS u α. Only terms coming from 2Re P P f and 4Re α 1 2 P A can a priori contribute since other terms are zero due to the symmetry of u α. The contribution of 2Re P P f is 2Re Π 1 Θ, P P f Φ α 2Re Φ α, P P f Π 1 Θ. We write Π 1 Θ as (α 1 2 γ 1 Γ 1 + Π 1 R)u α, where R and γ 1 are defined by (A.3) and (A.4) in Theorem A.1. This implies 2Re Π 1 Θ, P P f Φ α 2Re Φ α, P P f Π 1 Θ = 4Re (α 1 2 γ1 Γ 1 + Π 1 R)u α, P P f 2α 1 2 (Hf + Pf 2 ) 1 P P f Γ 1 u α = 8αRe γ 1 P P f Γ 1 u α, (H f + Pf 2 ) 1 P P f Γ 1 u α 8α 1 2 Re P uα P f Π 1 R, P u α (H f + Pf 2 ) 1 P f Γ 1 2αRe γ 1 Φ α 2 cα P u α 2 P f Π 1 Γ 1 = 2α Φ α 2 + O(α 4 ), (2.19) where in the last equality, we used P u α = O(α) and from (A.5) of Theorem A.1 that γ 1 1 = O(α) and Π 1 R R = O(α 3 2 ).

10 316 Jean-Marie Barbaroux and Semjon Vugalter The contribution of 4α 1 2 Re P.A is 4α 1 2 Re Π0 Θ, P A Φ α 4α 1 2 Re Φα, P A Π 2 Θ = 4α 1 2 Re P A + Π 0 Θ, Φ α 4α 1 2 Re 2α 1 2 P Pf (H f + P 2 f ) 1 Γ 1 u α, P A (αγ 2 Γ 2 + Π 2 R)u α = 4α 1 2 Re P A + Ω f u α, Φ α 8α 2 Re γ 2 P u α P f (H f + P 2 f ) 1 Γ 1, P u α A Γ 2 8αRe P u α P f (H f + P 2 f ) 1 Γ 1, P u α A Π 2 R = O(α 4 ), (2.20) where in the fourth inequality we used P u α = O(α), A Π 2 R c Π 2 R = O(α 3 2 ) from (A.5) in Theorem A.1, and P A + u α Ω f, Φ α = 0 from Lemma B.1. The estimates (2.19) and (2.20) yields α 1 2 Φα, HΘ + Θ, Hα 1 2 Φα = 2α Φ α 2 + O(α 4 ). (2.21) Step 4. We next estimate in (2.11) the cross terms involving Υ α and Θ = θ GS u α. As in the previous step, only terms coming from 2Re P P f and 4Re α 1 2 P A can a priori contribute since other terms are zero due to the symmetry of u α. The contribution of 2Re P P f is 2Re Π 1 Θ, P P f α 1 2 Υα 2Re α 1 2 Υα, P P f Θ = 4Re (α 1 2 γ1 Γ 1 + Π 1 R)u α, P P f α 1 2 Υα = 2αRe γ 1 2 P P f (H f + P 2 f ) 1 Γ 1 u α, Υ α 4α 1 2 Re P uα P f Π 1 R, Υ α = O(α 4 ) (2.22) where we used 2 P P f (H f + P 2 f ) 1 Γ 1 u α, Υ α = Φ α, Υ α = 0 due to Lemma B.1, and P u α = O(α), Υ α = O(α) and P f Π 1 R R = O(α 3 2 ) (Theorem A.1). The contribution of 4α 1 2 Re P A is 4α 1 2 Re Π0 Θ, P A α 1 2 Υα 4α 1 2 Re α 1 2 Υα, P A Π 2 Θ = 2αRe 2P A + u α Ω f, Υ α 4αRe Υ α, P A (αγ 2 Γ 2 + Π 2 R)u α = 2α Υ α 2 + O(α 4 ), (2.23) where we applied in the last equality P u α = O(α), Υ α = O(α) and A Π 2 R c R = O(α 3 2 ).

11 Contribution of the spin-zeeman term Equations (2.22) and (2.23) implies α 1 2 Υα, H Θ + Θ, H α 1 2 Υα = 2α Υ α 2 + O(α 4 ). (2.24) Step 5. Collecting all above estimates (2.12), (2.13), (2.14), (2.18), (2.21) and (2.24) yields Ψ trial, HΨ trial = (Σ 0 α2 4 ) Θ 2 α Φ α 2 α Υ α 2 + O(α 4 ) (2.25) To conclude the proof, we need to normalize the above expression. First note that Θ, α 1 2 (Φ α + Υ α ) = 0 due to orthogonality of u α and u α / x j (j = 1, 2, 3). Therefore since Φ α + Υ α = O(α log α 1 2 ). This yields Ψ trial 2 = Θ 2 + α Φ α + Υ α 2 Σ Ψtrial, H Ψ trial Ψ trial 2 = Θ 2 + O(α 3 log α ), = (Σ 0 α2 4 ) Θ 2 α Φ α 2 α Υ α 2 + O(α 4 ) Θ 2 + O(α 3 log α ) = (Σ 0 α2 4 ) α Φ α 2 α Υ α 2 + O(α 4 log α ), (2.26) where we used Θ 2 = 1 + O(α) (see Theorem A.1), Σ 0 = O(α), Φ α = O(α), and Υ α = O(α). To conclude the proof, it suffices to replace Φ α and Υ α by their expressions in Lemma B.2. A. ground state of T (0) To define the trial function Ψ trial in the proof of Theorem 2.1, we need some properties derived in [11] for the ground state of the self-energy operator with total momentum zero T (0). For convenience of the readers, we remind here these properties. Theorem A.1. Let Γ 1 := (H f + P 2 f ) 1 σ B + Ω f (A.1) and Γ 2 = (H f + P 2 f ) 1 ( σ B + Γ 1 + 2A + P f Γ 1 + A + A + Ω f ). (A.2)

12 318 Jean-Marie Barbaroux and Semjon Vugalter We have inf spec(t (0)) = α Γ α 2 ( 2 A Γ 1 2 Γ Γ 1 2 Γ 1 2) + O(α 3 ). In addition, let θ GS be the ground state of T (0) such that Π 0 θ GS = Ω f. Taking the.,. -orthonormal projections of θ GS along the vectors Γ 1 and Γ 2, and denoting by R the component in the.,. -orthogonal complement of their span, we get θ GS = Ω f + α 1 2 γ1 Γ 1 + αγ 2 Γ 2 + R (A.3) where for i = 1, 2 Γ i, R = 0 and Ω f, R = 0. (A.4) Then, we have γ 1 1 = O(α), γ 2 1 = O(α 1 2 ), R = O(α 3 2 ) and R = O(α). (A.5) B. Technical results Lemma B.1. For Φ α defined by (2.8) and Γ 1 defined by (A.1), we have for all α > 0 P A + u α Ω f, Φ α = 0 and Φ α, Υ α = 0. Proof. This is a straightforward computation. Lemma B.2. For Φ α defined by (2.8) and Υ α defined by (2.9), we have for all α > 0 Φ α 2 = 2α2 t 2 ζ 2 (t) 3π 0 (1 + t) 3 dt, and Υ α 2 = 2α2 ζ 2 (t) 3π t dt + O(α3 ). Proof. This is a straightforward computation using the definition of Φ α and Υ α, and the fact that (see e.g. [9]) k1 2 + k2 2 σ B + i ζ( k ) 0 Ω f = (k 1 +ik 2 )k 3 2π k 1 2 k 2 1 +k2 2, k ( k 2 +ik 1 ) k 2 1 +k2 2

13 Contribution of the spin-zeeman term and (H f + P 2 f ) 1 A + Ω f = ζ( k ) 2π k 1 2 ( k 2 + k ) k k2 2 k 2 + k 1k 3 k k 1 + k 2k 3 k (k 2 1 +k2 2 ) k. Acknowledgments J.-M. B. gratefully acknowledges financial support from Agence Nationale de la Recherche, via the projects HAM-MARK ANR-09-BLAN and ANR-12-JS He was also supported by BFHZ/CCUFB with the Franco-Bavarian cooperation project Mathematical models of radiative and relativistic effects in atoms and molecules. S. V. was supported by the DFG Project SFB TR Part of this work was done during the stay of S. V at the University of Toulon. References [1] V. Bach, T. Chen, J. Fröhlich and I.M. Sigal, The renormalized electron mass in non-relativistic QED., J. Funct. Anal., 243 (2007), [2] V. Bach, J. Fröhlich and A. Pizzo, Infrared-finite algorithms in QED: the groundstate of an atom interacting with the quantized radiation field, Comm. Math. Phys., 264, 1 (2006), [3] V. Bach, J. Fröhlich and I.M. Sigal, Mathematical theory of radiation, Found. Phys., 27, 2 (1997), [4] V. Bach, J. Fröhlich and I.M. Sigal, Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field, Comm. Math. Phys., 207, 2 (1999), [5] V. Bach, J. Fröhlich and I.M. Sigal, Quantum electrodynamics of confined relativistic particles, Adv. Math., 137, 2 (1998), [6] J.-M. Barbaroux, T. Chen and S.A. Vugalter, Binding conditions for atomic N-electron systems in non-relativistic QED., Ann. Henri Poincaré, 4 (2003), [7] J.-M. Barbaroux, T. Chen, V. Vougalter and S.A. Vugalter, On the ground state energy of the translation invariant Pauli-Fierz model, Proc. Amer. Math. Soc., 136 (2008), [8] J.-M. Barbaroux, T. Chen, V. Vougalter and S.A. Vugalter, Quantitative estimates on the binding energy for Hydrogen in non-relativistic QED., Ann. Henri Poincaré, 11 (2010), [9] J.-M. Barbaroux, H. Linde and S.A. Vugalter, Quantitative estimates on the enhanced binding for the Pauli-Fierz operator, J. Math. Phys., 46, 12 (2005), (11 pp).

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